Vasseur, Alexis F.2012-09-202017-05-112012-09-202017-05-112008-05http://hdl.handle.net/2152/17952textThe first part of this thesis is devoted to a regularity criterion for solutions of the Incompressible Navier-Stokes equations in terms of regularity of the solutions along the streamlines. More precisely, we prove that we can ensure the full regularity of a given suitable weak solution provided we have good control on the second derivative of the velocity along the direction of the streamlines of the fluid. In the second part of this thesis, we will show that the global regularity of a suitable weak solution u for the incompressible Navier-Stokes equations holds under the condition that [mathematical equation] is integrable in space time variables.electronicengCopyright is held by the author. Presentation of this material on the Libraries' web site by University Libraries, The University of Texas at Austin was made possible under a limited license grant from the author who has retained all copyrights in the works.Navier-Stokes equations--Numerical solutions